Problem 88 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{5}{6 a^{3}}+\frac{7}{8 a^{2}}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{20 + 21a}{24a^3}\).

Step 1: Determine the Common Denominator

The expressions have different denominators, namely \(6a^3\) and \(8a^2\). To combine them, we need a common denominator. The least common multiple (LCM) of \(6\) and \(8\) is \(24\), and for the variables, the highest power appearing is \(a^3\). So, the common denominator is \(24a^3\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite the first fraction \(\frac{5}{6a^3}\) as \(\frac{5 \times 4}{6a^3 \times 4} = \frac{20}{24a^3}\), and the second fraction \(\frac{7}{8a^2}\) as \(\frac{7 \times 3a}{8a^2 \times 3a} = \frac{21a}{24a^3}\).

Step 3: Combine the Fractions

With the common denominator, the expression becomes: \(\frac{20}{24a^3} + \frac{21a}{24a^3} = \frac{20 + 21a}{24a^3}\).

Step 4: Simplify the Result

The numerator \(20 + 21a\) cannot be simplified further with the denominator \(24a^3\), as they have no common factors other than 1. Therefore, the simplified result is \(\frac{20 + 21a}{24a^3}\).

Key Concepts

Least Common Multiple (LCM)

When dealing with algebraic fractions, especially when they have different denominators, finding a common base to work from is crucial. This is where the concept of the Least Common Multiple (LCM) comes in. The LCM is essentially the smallest multiple that two or more numbers (or algebraic expressions) share.

To find the LCM of two numbers first, we list out the multiples of each number. For the exercise at hand, with the denominators 6 and 8, we identify that the LCM is 24. Why 24? Because it's the smallest number that both 6 and 8 divide into perfectly.

• For the number 6: 6, 12, 18, 24...
• For the number 8: 8, 16, 24...

For algebraic terms like the variable ‘a’, take the highest power of the variable present in any of the denominators, which is a crucial detail. In our example, we have terms like \(a^3\) and \(a^2\); hence, we use \(a^3\) for the LCM so every part of the fractions can be accommodated uniformly. This approach to building a common ground allows the subsequent operation of combining fractions to proceed smoothly.

Combining Fractions

After establishing a common denominator using the LCM, the next step in handling algebraic fractions is to rewrite individual fractions so that they share the same denominator. By doing this, we align the fractions and make them ready for addition or subtraction.

To transform each fraction into one with the common denominator, we multiply both the numerator and the denominator by the same value. This multiplication doesn't change the value of the fractions but adjusts their form so they match in denominators. In our given problem:

• The fraction \(\frac{5}{6a^3}\) gets converted to \(\frac{20}{24a^3}\) once both the numerator and the denominator are multiplied by 4.
• Similarly, \(\frac{7}{8a^2}\) is transformed into \(\frac{21a}{24a^3}\). Here, both the numerator and denominator multiply by \(3a\) to reach uniformity.

With both fractions converted to a common base, adding them becomes straightforward. Combine the numerators on top of the shared denominator to produce a single, continuous fraction: \(\frac{20 + 21a}{24a^3}\). This approach leverages the establishment of a common ground, putting fractions in harmony with each other for smooth arithmetic operations.

Simplifying Expressions

Once the fractions are combined into a single expression, the final task is to simplify, if possible. Simplifying expressions means reducing them to their most basic form, where no further factoring or reduction is possible.

For the fraction \(\frac{20 + 21a}{24a^3}\), the numerator, \(20 + 21a\), does not have any common factors with the denominator, \(24a^3\), except for the number 1. This lack of shared factors means the expression is already in its simplest form. Simplifying usually involves factoring the numerator and denominator and canceling any common factors. However, in this case, we find none beyond the most basic unit.

• Check if the numerator can be factored - here, it cannot.
• Compare with the denominator's factors - \(24a^3\) is already factored, thus simplifying further isn't needed.

Therefore, the expression \(\frac{20 + 21a}{24a^3}\) is as simplified as it can get. Recognizing when an expression is at its basic form is just as important as performing calculations, easing mathematical manipulations and ensuring clarity.